%!TEX root = /Users/dawes/work/phd/dawes-phd-thesis/Dawes.tex
\chapter{All-Optical Switching} % (fold)
\label{cha:opt_switching}

An all-optical switch is a device that allows us to control of one beam of light with another. Beams of light do not interact when propagating in vacuum due to the linearity of Maxwell's equations, so all-optical switches rely on nonlinear interactions between light and matter. Two fundamental properties of a switch are that the device exhibit at least two distinguishable states, and that the device input and output are distinguishable.

\begin{figure}[htbp]
  \centering
    \includegraphics[width=6in]{Figures/switch_types.pdf}
  \caption[Types of all-optical switches]{Four generic types of all-optical switch. Each switch has two states, indicated by the two columns in the diagram. Each row illustrates one variation on the type of switch. a) The presence of a control beam changes the direction of a beam propagating through the medium. b) The control beam changes the direction of a beam generated within the medium. c) The control beam causes a beam propagating through the medium to be absorbed or scattered. d) The control beam prevents the generation of a beam within the medium.}
  \label{fig:switch_types}
\end{figure}

There are many possible configurations for all-optical switches, as shown in Fig.~\ref{fig:switch_types}. In general, all-optical switches can change the output power, direction, or state of polarization of a beam of light that is either propagating through a nonlinear medium or generated within the medium. Figure~\ref{fig:switch_types} illustrates devices that change the direction or power of either transmitted or generated beams. These are generic devices without specific implementation details. In order to clarify these configurations and introduce other concepts that are fundamental to all-optical switching, the following sections introduce two specific examples of all-optical switches. The first is based on an interferometer and relies on the nonlinear phase shift experienced by beams in one arm of the interferometer. The second uses a saturable absorber as the nonlinear medium, where a strong beam controls the absorption experienced by a weak beam.

\section{All-Optical Switching via Nonlinear Phase Shift} % (fold)
\label{sec:all_optical_switching_via_nonlinear_phase_shift}

One simple all-optical switch that has been demonstrated in a wide variety of  materials is based on the intensity-dependent refractive index of transparent nonlinear optical media. The intensity-dependent refractive index leads to a nonlinear phase shift experienced by a wave propagating through the medium. This effect can be used in all-optical switching by inserting such a medium in one arm of an interferometer.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/one_beam_phase_shift.pdf}
  \end{center}
  \caption{A beam traveling through a nonlinear medium experiences a phase shift $\phi_{\mathrm{nl}}$ that depends on the intensity of the beam.}
  \label{fig:one_beam_phase_shift}
\end{figure}

To describe the nonlinear phase shift, consider the situation illustrated in Fig.~\ref{fig:one_beam_phase_shift}. A strong beam of monochromatic light with complex field amplitude $E_c$, modifies its own propagation through a medium and exits with an accumulated nonlinear phase shift $\phi_{\mathrm{nl}}$. The total field within the medium is described by
\begin{equation}
  \label{eqn:in_field}
  \widetilde{E}(\mathbf{r},t)=E_c e^{-i\omega t}+\mathrm{c.c.}
\end{equation}
where $\omega$ is the frequency of the field and the complex amplitude $E_c$ contains the spatial dependence of the field $e^{i\mathrm{k_c\cdot r}}$.\footnote{I adopt a notation where the tilde indicates quantities that vary at optical frequencies.} The present treatment assumes that all fields have the same state of polarization, hence $\widetilde{E}$ is a scalar quantity. In general, a complete treatment considers both polarization states as described in Sec.~\ref{sec:polarization_instabilities}. The effect of the medium on the propagation of this field can be described in terms of the macroscopic polarization of the medium
\begin{equation}
  \label{eqn:pol}
  \widetilde{P}=\epsilon_0\chi \widetilde{E}
\end{equation}
where $\chi$ is the electric susceptibility of the medium, and $\epsilon_0$ is the permittivity of free space. In nonlinear optics, the optical response can often be described by an expansion of Eq.~(\ref{eqn:pol}) in powers of the incident field $\widetilde{E}$
\begin{equation}
  \widetilde{P}=\epsilon_0\left(\chi^{(1)} \widetilde{E} + \chi^{(2)} \widetilde{E}^2 + \chi^{(3)} \widetilde{E}^3 + \ldots\right).
\end{equation}
For isotropic media, such as an atomic vapor, $\chi^{(2)}=0$, so the lowest-order nonlinear term in the polarization is described by $\chi^{(3)}$ \cite{Boyd_2002aa}. The linear and nonlinear parts of the polarization can be considered separately as
\begin{equation}
  \widetilde{P}=\widetilde{P}^{(1)}+\widetilde{P}^\mathrm{NL}
\end{equation}
where $\widetilde{P}^{(1)}=\epsilon_0 \chi^{(1)} \widetilde{E}$ and $\widetilde{P}^\mathrm{NL}=\epsilon_0 \chi^{(3)} \widetilde{E}^3$. For simplicity in treating the large number of terms that contribute to the polarization, $\widetilde{P}^\mathrm{NL}$ can be expressed in terms of the amplitudes $P(\omega_n)$
\begin{equation}
  \widetilde{P}^\mathrm{NL}=\sum_n P(\omega_n)e^{-i\omega_n t}.
\end{equation}

For the present case of a single beam, where the total incident field is given by Eq.~(\ref{eqn:in_field}), the polarization component of interest, $P(\omega)$, is the one with the same frequency dependence as the incident field. This component of the polarization describes the effect of the nonlinear medium on the incident field with frequency $\omega$.\footnote{There are, of course, many other components of the polarization, each with different frequency dependence including all sum and difference frequencies within $\widetilde{E}^3$.} From Eq.~(\ref{eqn:in_field}), $\widetilde{E}^3$ contributes three terms to $P(\omega=\omega+\omega-\omega)$
\begin{equation}
  \label{eqn:perm}
  E_c E_c E_c^*,\,E_c E_c^* E_c,\,E_c^* E_c E_c
\end{equation}
where, again, the complex conjugate amplitude $E_c^*$ is associated with frequency $-\omega$. The incident field [Eq.~(\ref{eqn:in_field})] thus leads to a nonlinear polarization of the medium given by
\begin{equation}
  \label{eqn:self_phase_P}
  P^\mathrm{NL}=3\epsilon_0\chithree|E_c|^2 E_c.
\end{equation}
The numerical factor of 3 appearing in Eq.~(\ref{eqn:self_phase_P}) is known as the degeneracy factor and is given by the number of distinct permutations [Eq.~(\ref{eqn:perm})] of the frequencies contributing to $P^\mathrm{NL}$. To determine the effect of this nonlinear polarization on the refractive index of the medium, an effective susceptibility $\chi_\mathrm{eff}$ can be defined by
\begin{equation}
  P=\epsilon_0\chi_\mathrm{eff}E_c,
\end{equation}
such that
\begin{equation}
  \chi_\mathrm{eff}=\chi^{(1)}+3\chi^{(3)}|E_c|^2,
\end{equation}
where $\chi^{(1)}$ is the linear susceptibility of the medium. The index of refraction can be written in terms of the effective susceptibility as follows
\begin{equation}
 \label{eqn:index_via_chi} n=\left(1+\chi_\mathrm{eff}\right)^{1/2}=n_0\left(1+\frac{3\chi^{(3)}|E_c|^2}{n_0^2}\right)^{1/2}\simeq n_0+\frac{3\chi^{(3)}|E_c|^2}{2 n_0},
\end{equation}
where $n_0=\sqrt{1+\chi^{(1)}}$ is the background, or linear, refractive index of the medium. For intensity $I_c=2\epsilon_0 n_0 c |E_c|^2$, the index of refraction is then given by \cite{Boyd_2002aa}
\begin{equation}
	n=n_0+n_2I,
\end{equation}
with
\begin{equation}
  \label{eqn:intensity_index}
  n_2=\frac{3}{4 \epsilon_0 n_0^2 c}\chithree.
\end{equation}
where $c$ is the speed of light in vacuum and I have assumed $n_2<<n_0$ in order to ignore higher-order terms in the expansion of Eq.~(\ref{eqn:index_via_chi}). The phase acquired by the beam due to this nonlinear index of refraction is given by
\begin{equation}
  \phi_\mathrm{nl}=k_\mathrm{nl}L=\frac{n_2 I \omega L}{c},
\end{equation}
where $k_\mathrm{nl}$ is the nonlinear portion of the wavevector $k=n(\omega/c)L$, $L$ is the length of the medium, $\omega$ is the optical frequency, and $c$ is the speed of light in vacuum.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/two_beam_phase_shift.pdf}
  \end{center}
  \caption{A strong beam effects the propagation of a weak beam by inducing a nonlinear phase shift, $\phi_{\mathrm{nl}}$.}
  \label{fig:two_beam_phase_shift}
\end{figure}

Another configuration for observing the nonlinear phase shift is shown in Fig.~\ref{fig:two_beam_phase_shift}, where a strong beam of light of amplitude $E_c$ modifies the propagation of a weak beam of amplitude $E_s$. In this situation, the effect of the strong wave on the weak wave is twice as strong as the effect of the strong wave on itself. The total field for the two-beam case is
\begin{equation}
  \label{eqn:2in_field}
  \widetilde{E}(\mathbf{r},t)=E_c e^{-i\omega t}+E_s e^{-i\omega t}+\mathrm{c.c.}
\end{equation}
As above, the total field Eq.~(\ref{eqn:2in_field}) contributes to the polarization $P(\omega=\omega+\omega-\omega)$. Now there are 6 terms in $\widetilde{E}^3$ that have frequency dependence $\omega$
\begin{equation}
  \label{eqn:perm2}
  E_c E_c^* E_s,\,E_c^* E_c E_s,\,E_c E_s E_c^*,\,E_c^* E_s E_c,\,E_s E_c E_c^*,\,E_s E_c^* E_c.
\end{equation}
The nonlinear polarization of the medium that describes the effect of the control beam $E_c$ on the signal beam $E_s$ is thus given by
\begin{equation}
  P^\mathrm{NL}=6\epsilon_0\chithree|E_c|^2 E_s,
\end{equation}
where the degeneracy factor (6) is twice that of Eq.~(\ref{eqn:self_phase_P}). The extra degeneracy arises because the fields $E_c(\omega)$ and $E_s(\omega)$, despite having the same frequency, are distinguishable by propagation direction. Following a derivation along the lines of the single-beam case described above, the intensity-dependent index of refraction experienced by the weak wave due to the presence of the strong wave is given by
\begin{equation}
  \label{eqn:weak_wave_n2}
  n=n_0+2n_2 I_c,
\end{equation}
where the intensity of the control beam is given by $I_c=2 \epsilon_0 n_0 c|E_c|^2$, \emph{i.e.}, it is the intensity of the strong wave that leads to a phase shift experienced by the weak one.

If the nonlinear medium is inserted into one arm of an interferometer, the strong wave of Fig.~\ref{fig:two_beam_phase_shift} can be used as a control beam, and impart a phase shift in one arm of the interferometer, thus serving to control which output port sees constructive or destructive interference. An illustration of this all-optical switching scheme is shown in Fig.~\ref{fig:phase_switch}.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/phase_switch.pdf}
  \end{center}
  \caption[Nonlinear phase shift in a Mach-Zehnder interferometer can be used as an all-optical switch.]{In a Mach-Zehnder interferometer, a beamsplitter (BS) splits and then combines an incident beam. Interference is detected in the two output beamsplitter ports as variations in intensity depending on the relative phase shift between the two beam paths. A nonlinear medium is inserted in one path of the interferometer to make an all-optical switch.}
  \label{fig:phase_switch}
\end{figure}

The following is a derivation of the interferometer output for a given nonlinear phase shift. The signal field ($E_s$) enters one port of the first beamsplitter, and the control field ($E_c$) enters medium at an angle relative to the signal field. In one arm of the Mach-Zehnder interferometer, the weak wave experiences  a nonlinear refractive index, $n=n_0+2n_2I$. The other arm of the interferometer is linear, where the beam only experiences the usual phase shift due to propagation through a medium of index $n_0$. The output of the interferometer can be measured at ports 1 and 2, and is described as follows.

We first assume the following beamsplitter relations:\footnote{This form of the beam-splitter relation ensures that the transfer characteristics obey the Stokes relations.}
\begin{align}
  r&=i\sqrt{R}\\
  t&=\sqrt{T}\\
  R+T&=1,
\end{align}
where $r$ ($t$) is the reflection (transmission) coefficient for the field, and $R$ ($T$) is the intensity reflection (transmission) coefficient.
The output field, at port 1, can then be written as:
\begin{align}
  E_1=E_s(rt e^{i k L}+rt e^{i k_{nl} L}),
\end{align}
with $k=n_0 \omega / c$, and $k_{nl}=(n_0 + 2n_2 I)(\omega / c)$ which, dropping the common phase term $\exp[i n_0 (\omega/c) L]$ gives:
\begin{align}
  E_1=E_s(rt+rt e^{i \phi_{nl}}),
\end{align}
where I have defined
\begin{equation}
  \label{eqn:phi_nl}
  \phi_{nl}=2\frac{\omega}{c}n_2IL,
\end{equation} 
to be the nonlinear phase experienced in one arm of the interferometer. This gives the output intensity at port 1:
\begin{align}
  |E_1|^2&=|E_s|^2(rt+rt e^{i \phi_{nl}})(r^*t+r^*t e^{-i \phi_{nl}})\nonumber\\
  &=|E_s|^2(|r|^2|t|^2)(1+e^{i \phi_{nl}})(1+e^{-i \phi_{nl}})\nonumber\\
  &=2|E_s|^2RT(1+\cos\phi_{nl}).
\end{align}
Similarly, the intensity at output port 2 is given by:
\begin{align}
  |E_2|^2&=|E_s|^2(r^2+t^2 e^{i \phi_{nl}})((r^*)^2+t^2 e^{i \phi_{nl}})\nonumber\\
  &=|E_s|^2(r^4 + r^2t^2e^{-i \phi_{nl}} + (r^*)^2t^2 e^{i \phi_{nl}} + t^4)\nonumber\\
  &=|E_s|^2(R^2 + T^2 - 2RT\cos\phi_{nl}),
\end{align}
where $|E_1|^2+|E_2|^2=|E_s|^2$.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/interferometer.pdf}
  \end{center}
  \caption[Interferometer output as a function of relative phase difference.]{The output of the interferometer shown in Fig.~\ref{fig:phase_switch}. The solid line is the intensity of output port 1 as a function of phase shift $\phi_\mathrm{nl}$ in units of $|E_s|^2$. Similarly, the dashed line is the intensity at output port 2. For full-contrast switching, a phase shift of $\pi$ is required between the two arms of the interferometer. For this figure, $R=T=1/2$ as appropriate for 50/50 beamsplitters.}
  \label{fig:interferometer_out}
\end{figure}

The functional form of the output intensities clearly demonstrate the sinusoidal variation of the output intensity with $\phi_{nl}$. Figure~\ref{fig:interferometer_out} plots the output intensities at output 1 (solid curve), and output 2 (dashed curve) of the interferometer. The sinusoidal variation with phase is clear, as is the fact that the highest contrast between the two output states is observed when $\phi_{nl}=\pi$.

To illustrate the use of this device, we consider what conditions lead to a nonlinear phase shift of $\phi_{nl}=\pi$, which is required for high-contrast switching. Silica glass is a readily available material that exhibits an intensity-dependent refractive index. The nonlinear susceptibility for silica is $\chithree=3.2\times10^{-16}$~(cm$^2$/W), which is not particularly large. Given Eqs.~(\ref{eqn:weak_wave_n2}) and (\ref{eqn:phi_nl}), an intensity of $3.9\times10^{10}$~W/cm$^2$ is required for a $\pi$ phase shift if the length of the nonlinear medium is only 5~cm. However, as silica glass can be drawn into optical fiber, the medium length can be made large enough to observe a significant effect with typical laser powers. Fifty kilometers of silica glass requires an intensity of 39 kW/cm$^2$ to achieve a full $\pi$ phase shift. Light in a silica glass fiber 4 microns in diameter can reach this intensity with a few milliwatts of optical power.

The preceding discussion shows that an all-optical switch based on the nonlinear phase shift requires a nonlinear phase shift on the order of $\pi$ radians. I have used realistic parameters to model this prototypical device to illustrate another point: the nonlinearity leading to a phase shift is not very strong, and the total intensity required to reach the phase shift necessary for all-optical switching in this configuration can be significant.

% section all_optical_switching_via_nonlinear_phase_shift (end)

\section{All-Optical Switching via Saturated Absorption} % (fold)
\label{sec:all_optical_switching_via_saturation_of_a_two_level_atom}

A second type of all optical switch relies on the properties of a \emph{saturable absorber}. The absorption experienced by a wave propagating through a homogeneously-broadened medium that exhibits saturable absorption depends on the intensity, and decreases for increasing intensity following the relation \cite{Boyd_2002aa}
\begin{equation}
  \label{eqn:saturable_absorption} 
	\alpha=\frac{\alpha_0}{1+I/I_s},
\end{equation}
where $\alpha_0$ is the absorption coefficient experienced by a weak field, $I_s$ is a constant of the medium known as the saturation intensity defined such that when the intensity equals the saturation intensity, $I=I_s$, the absorption $\alpha$ is reduced to half the weak-beam value $\alpha(I=I_s)=\alpha_0/2$.\footnote{Most materials exhibit saturable absorption, although some only for very high intensities. Some organic dye solutions and doped solids, however, exhibit reverse saturable absorption where the transmission decreases for larger intensity.}

Using Beer's law for absorption, it is straightforward to describe the attenuation experienced by a wave propagating through a medium with a uniform absorption coefficient $\alpha_0$:
\begin{equation}
  \label{eqn:beers} 
  I(z)=I_0 e^{-\alpha_0 z},
\end{equation}
where $I_0$ is the intensity before entering the medium. We next modify this attenuation of the field to account for the fact that $\alpha$ depends on the intensity, which will allow me to describe the saturation resulting from an intense wave propagating in the medium. The goal of this derivation is to describe the effect of a strong saturating beam on the absorption experienced by a weak signal beam. This situation is illustrated in Fig.~\ref{fig:saturation_switch}.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/saturation_switch.pdf}
  \end{center}
  \caption[An all-optical switch based on saturable absorption.]{An all-optical switch based on saturable absorption. (a) The absorption experienced by the signal wave, is reduced (b) by the presence of a strong control wave that saturates the medium. The attenuation of the signal is thereby controlled by the saturating control wave.}
  \label{fig:saturation_switch}
\end{figure}

\newcommand{\isat}{\ensuremath{I_{\mathrm{sat}}}}
An intensity-dependent absorption coefficient modifies the usual Beer's law absorption Eq.~(\ref{eqn:beers}) as follows. The intensity of a strong wave, which I will call the control wave $I_c(z)$ obeys
\begin{equation}
  \frac{dI_c}{dz}=-\alpha_c I_c,
\end{equation}
where we have saturable absorption given by Eq.~(\ref{eqn:saturable_absorption}), which leads to
\begin{equation}
  \label{eqn:intensity}
  \frac{dI_c}{dz}=-\frac{\alpha_0}{1+I_c/\isat} I_c.
\end{equation}
Equation~(\ref{eqn:intensity}) can be integrated to yield
\begin{equation}
  \frac{I_c}{\isat}+\ln{(I_c)}=-\alpha_0 z + A,
\end{equation}
where $A$ is a constant of integration. Exponentiation of both sides reveals that the intensity follows the form of the Lambert W function\footnote{The Lambert W function is the inverse of the function $f(w)=w e^{w}.$}
\begin{equation}
  I_c(z) = \isat \operatorname{W}\left(\frac{A}{\isat} e^{-\alpha_0 z}\right).
\end{equation}
Finally, $A$ can been replaced with $I_{c,0} e^{I_{c,0}/\isat}$ by solving for the initial conditions, $I_c(0)=I_{c,0}$, doing so gives
\begin{equation}
  \label{eqn:control_beam} 
  I_c(z) = \isat \operatorname{W}\left(\frac{I_{c,0}}{\isat} e^{\frac{I_{c,0}}{\isat}-\alpha_0 z}\right).
\end{equation}

Next, we consider the effect of this strong saturating wave on the transmission of a weak wave propagating through the medium with intensity $I_w(z)$. This situation is illustrated in Fig.~\ref{fig:saturation_switch}(b). For simplicity, I will ignore the intensity grating formed by the interference of the strong and weak wave. The effects of this interference and the associated wave-mixing effects are described in the next chapter. Additionally, I assume $I_w(z)<<\isat<I_c(z)$, or that the weak beam is weak relative to the saturation intensity and the intensity of the control beam will typically be above the saturation intensity. These assumptions give the following expression for the absorption of the weak wave as a function of $z$
\begin{equation}
  \alpha_w(z)=\frac{\alpha_0}{1+\frac{I_c(z)}{\isat}},
\end{equation}
where the expression for $I_c(z)$ is given in Eq.~(\ref{eqn:control_beam}). The intensity of the weak beam is then given by
\begin{equation}
  \label{eqn:weak_wave_eqn}
  \frac{dI_w}{dz}=-\alpha_w(z) I_w.
\end{equation}
Which can be integrated to yield
\begin{equation}
  \label{eqn:abs_switch}
  I_w(z)=I_{w,0}\frac{ \operatorname{W} \left(\xi e^{\xi-z\alpha_0}\right)} {\operatorname{W} \left(\xi e^\xi\right)}
\end{equation}
with $\xi=I_{c,0}/\isat$.

To evaluate the response of the absorptive switch, we first examine the behavior of Eq.~(\ref{eqn:abs_switch}). Consider the transmission of the absorbing system, defined as $T=I_w/I_{w,0}$. For various initial intensities $I_{w,0}$, the transmission is plotted as a function of optical depth, $\alpha_0 L$ where $L$ is the length of the medium, in Fig.~\ref{fig:saturation_plot}. We interpret this plot as follows, a weak signal beam propagating alone experiences the largest amount of absorption, as shown by the lower solid line. A control beam with intensity equal to the saturation intensity causes the weak signal beam to experience less absorption (indicated by the dotted line). If the control beam intensity is increased to $2\isat$, the signal beam experiences even less absorption (upper solid curve).  Thus, in order to make a switch with large contrast between the on and off states, it is clear from Fig.~\ref{fig:saturation_plot} that the control beam intensity must be on the order of the saturation intensity. Furthermore, we see that the ideal optical depth corresponds to where there is the largest difference between the saturated absorption and the unsaturated absorption. It can be shown that this optimum depth is where $\alpha_0 L=1.3$. Examining Fig.~\ref{fig:saturation_plot} by eye confirms that, at $\alpha_0 L=1.3$, the saturated and unsaturated (dotted and lower-solid curves respectively) are further apart than for any other value of $\alpha_0 L$. At the optimum optical depth, $\alpha_0 L=1.3$, the transmission of a weak beam through an unsaturated medium is 27\%. In the presence of a strong beam with $I_{c,0}=I_\mathrm{sat}$, the transmission increases to 47\%.

\begin{figure}[htbp]
  \centering
    \includegraphics[scale=1]{Figures/AbsorptiveSwitch.pdf}
  \caption[Transmission through a saturable absorber for various input intensities]{Transmission through a saturable absorber for various input intensities. The three curves are $I_{c,0}=0.0001,1,2$ from bottom to top.}
  \label{fig:saturation_plot}
\end{figure}

In order to realize the switch proposed in Fig.~\ref{fig:saturation_switch}, the control beam intensity must be sufficient to saturate the atomic response (\emph{i.e.}  $I_{c,0}=\isat$), but we have yet to quantify $\isat$ for a generic medium. Saturation of a two-level system corresponds to moving a significant amount of the atomic population from the ground state to the excited state. In order to maintain population in the excited state, one photon must be incident on each atom per excited state lifetime. Quantitatively, this condition is \cite{Siegman_1986aa}
\begin{equation}
  \label{eqn:saturation_intensity}
	I_\mathrm{sat}=\frac{\hbar \omega}{\sigma \tau_\mathrm{sp}},
\end{equation}
where $\tau_\mathrm{sp}$ is the excited state lifetime, $\hbar\omega$ is the photon energy, and $\sigma$ is the atomic cross section. In a material undergoing optical pumping, the relevant time scale is instead the ground-state lifetime $\tau_g$. In an optically-pumped material, the population is redistributed in time $\tau_g$, hence to maintain saturation, one photon must be incident on each atom per $\tau_g$.

The cross section can be interpreted as the effective area of an atom for absorbing radiation from an incident beam of light. For a pair of isolated levels, with incident light tuned to resonance, without collisional or Doppler broadening, and assuming all atoms are optimally aligned to interact with the field, the cross section has a maximum value of \cite{Boyd_2002aa, Siegman_1986aa}
\begin{equation}
  \label{eqn:sigma}
	\sigma_\mathrm{max}=\frac{3\lambda^2}{2\pi},
\end{equation}
where $\lambda$ is the wavelength of incident light and the factor of three is replaced with unity for the case of randomly-aligned atomic dipoles \cite{Siegman_1986aa}.

The switch response plotted in Fig.~\ref{fig:saturation_plot} can now be evaluated for realistic conditions. A typical cloud of cold-trapped rubidium atoms satisfies the requirements for maximizing $\sigma$. The saturation intensity is determined using Eq.~(\ref{eqn:saturation_intensity}) and the parameters for the rubidium $D_2$ transition. I find $I_\mathrm{sat}=3$~mW/cm$^2$ for $\tau_\mathrm{sp}=25$~ns and $\lambda=780$~nm. Thus, for a beam with radius 2~mm, an optical power of 0.4~mW would be required to actuate a MOT-based switch, and hence change the transmission of a weak beam from 27\% to 47\%. Of course, one limitation of the saturation-based switch is that the weak beam must be weak relative to the saturation intensity. In the MOT case described here, the saturation intensity corresponds to a 2~mm diameter beam with $\sim$80~$\mu$W of power, hence the power of the signal beam must be much less than 80~$\mu$W, and therefore, much less than the power of the control beam.

Rather than evaluate these equations for every saturation-based switch, it is more convenient to establish a metric for comparing different all-optical switches having different geometries. The fact that the maximum atomic cross section depends only on $\lambda$ leads to one such metric. To make comparisons between all-optical switches of various sizes, we quantify the energy density of the control field in units of photons per $\lambda^2/(2\pi)$. The motivation for this definition is twofold: first, as shown above, the maximum absorption cross section of a randomly-oriented two-level atom is given by $\lambda^2/(2\pi)$. Second, the minimum transverse size of an optical beam is set by the diffraction limit to be on the order of $\lambda^2$. Thus, in principle, a larger device that operates at $n$ photons per $\lambda^2/(2\pi)$ can be scaled to the diffraction limit and operate with only $n$ photons.

One caveat in using the energy density for comparison between various all-optical devices is that this metric does not have an implicit unit of time. Clearly one photon per \lambdasquared~per year is a very different input from one photon per \lambdasquared~per picosecond. It is thus necessary to include the switch response time to accurately compare the sensitivity of devices using the energy density metric. The switch response time is then defined such that the switch can be actuated if the specified energy density is applied for at least the duration of the response time. This caveat arises from the motivation for defining the switching energy density, which is simply to compare a large-scale all-optical switch to a device that is geometrically optimized to operate at the limit of small optical beams: the diffraction limit.

The origin of the switching energy density metric is the calculation of the energy density required to achieve a notable decrease in the amount of absorption experienced in an ensemble of two-level atoms. A decrease in absorption occurs in an ensemble of two-level atoms if the control beam has sufficient intensity to maintain a significant fraction of the atoms in their excited state. From Eq.~(\ref{eqn:saturation_intensity}) and Eq.~(\ref{eqn:sigma}), we see this is achieved by having an input beam at the saturation intensity, or an energy density that corresponds to approximately one photon per \lambdasquared. Energy of this density must be applied for at least the lifetime of the excited state. Otherwise saturation will not occur. The assumption that saturation of the atomic transition is required for observing high-contrast all-optical switching led to the early conclusion that all-optical switches must operate with at least one photon per \lambdasquared ~\cite{Keyes_1970aa}, a point that is discussed in more detail in Chapter~\ref{cha:switch}.

In terms of the practical nature of the example switch based on saturable absorption, the primary drawback is the requirement that the signal beam must be very weak in comparison to the control beam. A strong signal beam would saturate the absorption itself and be transmitted regardless of the control beam. As I will discuss in the next Section, the requirement of a strong control beam limits the range of applications for a switch based on saturable absorption.

I have presented two prototypical devices that can be used for all-optical switching that use two different nonlinear mechanisms to achieve light-by-light control. The primary purpose of discussing these examples was to introduce the fundamental ideas of nonlinear phase shift, energy density, and absorption cross section. Next, I step back and present an overview of the general requirements for a practical all optical switch. The final section in this chapter is a review of recent progress in the field of all-optical switching.

% section all_optical_switching_via_saturation_of_a_two_level_atom (end)

\section{All-Optical Switching: Overview} % (fold)
\label{sec:switching_overview}

Switches can be used in two classes of applications: information networks and computing systems. In each of these applications, information can be stored in either classical or quantum degrees of freedom. Hence the requirements for a device vary depending on the intended application.

Classical, all-optical networks require switches to reliably redirect or gate a signal depending on the presence of a control field at the device input. Ideally the switch shows large contrast between on and off output levels and can be actuated by low input powers. If the network carries quantum information, the switch must be triggered by an input field containing only a single quanta (photon). Additionally, in the quantum case, the quantum state of the transmitted signal field must be preserved.

If a switch is to be used as a logic element in a classical computing system, it must have the following characteristics: input-output isolation, cascadability, and signal level restoration \cite{Keyes_2006aa}. Input-output isolation prohibits the device output from having back-action on the device input. Cascadability requires that a device output have sufficient power to drive the input of at least two identical devices. Signal level restoration occurs in any device that outputs a standard signal level in response to a wide range of input levels. That is, variations in the input level do not cause variations in the output level. Switching devices that satisfy these requirements are considered scalable devices, \emph{i.e.}, the properties of the individual device are suitable for scaling from a one device to a network of many devices.

While scalability describes important properties of a switching device, sensitivity provides one way to quantify its performance. A highly sensitive all-optical switch can be actuated by a very weak optical field. Typical metrics for quantifying sensitivity are: the input switching energy (in Joules), the input switching energy density (in photons per $\sigma=\lambda^2/2\pi$) \cite{Harris_1998aa, Keyes_1970aa}, and the total number of photons in the input switching pulse.

One may not expect a single device to satisfy all of the requirements for these different applications. For example, a switch operating as a logic element should output a standard level that is insensitive to input fluctuations. This may be at odds with quantum switch operation where the device must preserve the quantum state of the signal field. An interesting question arises from these requirements: What happens when a classical switch is made sensitive enough to respond to a single photon? Reaching the level of single photon sensitivity has been the goal of a large body of recent work that is reviewed below.

% section intro_to_switching (end)

\section{Previous Research on Low-Light-Level Switching}
\label{sec:previous_switches}

Two primary approaches to low-light-level switching have emerged, both of which seek to  to increase the strength of the nonlinear coupling between light and matter. The first method uses fields and atoms confined within, and strongly coupled to, a high-finesse optical cavity. The second method uses traveling waves that induce quantum interference within an optical medium and greatly enhance the effects of light on matter.

%\subsubsection{Cavities}

Cavity quantum-electrodynamic (QED) systems offer very high sensitivity by decreasing the number of photons required to saturate the response of an atom that is strongly coupled to a mode of the cavity. For small-mode-volume, high-finesse cavities containing a strongly-coupled atom, even a single cavity photon can affect measurable change in the cavity transmission. Among many remarkable experiments, strongly-coupled cavity QED systems have shown nonlinear optical response to fields with much less than a single cavity photon \cite{Hood_1998aa}, and have also demonstrated the photon blockade effect where the arrival and absorption of one photon prevents subsequent absorption of a second photon \cite{Birnbaum_2005aa}. Both of these observations imply the possible application of cavity QED systems as all-optical switching devices. 

In the strong-coupling regime, the rate of coherent, reversible evolution dominates the decoherence rates for the atom-cavity system. Working in this regime, Hood \etal measured the transmission of a 10~pW probe beam through a cavity with linewidth $\kappa=40$~MHz (cavity lifetime 25~ns) while cold caesium atoms were dropped through the cavity mode. When an atom is present in the cavity mode, the effect of a single cavity photon (on average), is an order of  magnitude increase in cavity transmission. For 10 cavity photons, the nonlinear optical response is saturated and almost complete transmission is observed \cite{Hood_1998aa}. Operating with a cavity mode waist of 15~$\mu$m, a single input photon ($\lambda=852$~nm) represents a switching energy density of $\sim$$10^{-4}$~photons/$\sigma$ and a total input energy of $\sim$$10^{-19}$~J. As discussed in later sections, other systems have only recently operated at such high sensitivity levels.

In a similar system, Birnbaum \etal \cite{Birnbaum_2005aa} observe an effect known as photon-blockade in analogy to the Coulomb-blockade effect observed in semiconductor charge transport. Photon-blockade in cavity QED systems exploits the anharmonic splitting of atomic energy levels for atoms strongly coupled to a high-finesse cavity. This level splitting allows the absorption of a single photon to prevent subsequent absorption of a second photon. For the probe frequency used in these experiments, the single-photon process is resonant with the lowest excited dressed-state of the atom-cavity system while the two-photon absorption process is suppressed. With an average of 0.21~photons in the cavity (mode waist $w=23.4~\mu$m), the sensitivity of this system is $\sim$$10^{-5}$~photons/$\sigma$, comparable to the lowest reported to date \cite{Zhang_2007aa}.

Cavity QED systems offer an extremely high sensitivity and are currently a leading candidate for nodes in a photon-based quantum-information-network. Although a single two-level atom in free space exhibits an effect similar to photon blockade: once excited it cannot immediately absorb a second photon, interactions between single photons and single atoms in free space are exceedingly difficult to control. Hence one major achievement of cavity QED is to make the single-photon, single-atom regime accessible, stable, and repeatable. The primary drawback to integrating cavity-QED-based devices into switching networks is that the cavity system is designed to operate in a single field mode, which limits the number of input and output channels to one channel per polarization. For the photon-blockade effect, controlling the transmission of later photons requires the presence of an initial photon in the same mode. One must then discriminate between signal photons and control photons in some way other than by input mode (such as by polarization). 

Of the scalability requirements for an all-optical switch, cavity QED systems do not satisfy cascadability. While coupling light into and out of an atom contained in the cavity is very efficient, all input and output signals are coupled strongly. Thus a weak input cannot have an effect on the transmission of a strong signal because the strong signal itself is coupled to the atom. The input and the output must have similar powers otherwise the stronger beam completely overrides the effect of the weaker beam. While well suited for interacting with single photons, cavity QED systems are not designed to allow single photon inputs to control strong, many-photon outputs.

A different technique for all-optical switching in cavities relies on creating and controlling cavity solitons. Cavity solitons are spatially localized structures appearing as intensity peaks in the field emitted by a nonlinear microresonator. The most recent experiments on all-optical switching with cavity solitons use vertical cavity surface emitting lasers (VCSELs) as the nonlinear cavity \cite{Hachair_2005aa}. A VCSEL can be prepared for cavity solitons by injecting a wide holding beam along the cavity axis the cavity. A narrow beam superimposed on the larger holding beam and traveling through the laser cavity serves as a write beam that induces a cavity soliton. Typically, the solitons persist until the original holding beam is turned off, hence this system naturally serves as a pixel-based optical memory, where solitons are written to and stored in the cavity field.

Cavity solitons in VCSEL systems are well-suited to applications in optical networks. The solitons function as pixels and can be addressed individually, alleviating the single-channel limitation of cavity QED systems. Lower sensitivity is the primary limitation of the cavity soliton systems. Although the soliton turn-on is very fast (500-800~ps), typical powers for the hold and write beams are 8~mW and 150~$\mu$W respectively. The lowest reported write beam power is 10~$\mu$W for a holding beam of 27~mW suggesting a compromise can be made to lower the required write power by increasing the hold beam power. Injecting 10~$\mu$W for 500~ps corresponds to an input pulse containing $\sim$24,000~photons, and a switching energy density of $\sim$140 photons/$\sigma$ (write beam diameter 10~$\mu$m, $\lambda=960-980$~nm). Extinguishing solitons (as would be required for operation as a switch) involves either cycling the holding beam or injecting a second ``write'' pulse out of phase to erase the soliton. Either modification adds a slight complication to the device. 

In terms of the scalability criteria discussed above, cavity soliton devices satisfy the requirements of signal level restoration because the soliton intensity stabilizes to a consistent level for a large range of write beam intensities. Unlike cavity QED devices, large-area cavity soliton systems are inherently multi-mode and can be made very parallel with each potential soliton location serving as an isolated input/output channel. Cascadability may present a challenge for cavity soliton devices, however. Each cavity soliton would have to emit enough power to seed solitons in two or more subsequent devices and imaging or other beam shaping techniques may be required to properly image the output soliton into a second cavity. To the best of my knowledge this problem has not been addressed in the literature.

%\subsubsection{Quantum Interference}

In contrast to cavity systems, traveling wave approaches can operate with multi-mode optical fields and also achieve few-photon sensitivity. Recent progress in traveling-wave low-light-level nonlinear optics has been made through the techniques of electro-magnetically induced transparency (EIT) \cite{Harris_1997aa,Schmidt_1996aa,Zibrov_1999aa,Braje_2003aa, Chen_2005aa,Zhang_2007aa}. As an example, Harris and Yamamoto \cite{Harris_1998aa} proposed a switching scheme using the strong nonlinearities that exist in specific states of four-level atoms where, in the ideal limit, a single photon at one frequency causes the absorption of light at another frequency. To achieve the lowest switching energies, the narrowest possible atomic resonances are required which is the main challenge in implementing this proposal. Suitably narrow resonances can be obtained in complex experimental environments, for example, trapped cold atoms \cite{Harris_1999aa,Yan_2001aa,Braje_2003aa,Chen_2005aa,Zhang_2007aa}. 

Using trapped cold atoms, and the Harris-Yamamoto scheme, Braje \etal \cite{Braje_2003aa} first observed all-optical switching in an EIT medium with an input energy density of $\sim$23~photons/$\sigma$. Subsequent work by Chen \etal \cite{Chen_2005aa} confirmed that the EIT switching scheme operates at the 1~photon/$\sigma$ level. Using a modified version of the Harris-Yamamoto scheme with an additional EIT coupling field,  Zhang \etal \cite{Zhang_2007aa} recently observed switching with $\sim $20 photons ($10^{-12}$~W for $\tau_r=$0.7~$\mu$s with a 0.5~mm beam diameter) corresponding to $10^{-5}$~photons/$\sigma$. This is the highest all-optical switching sensitivity reported to date.

Althought EIT switches are very sensitive, the input and output fields are necessarily of the same strength so the requirements for cascadability are not met. The other feature of scalable devices missing from these proposals is signal level restoration. The output level for EIT based switches is a monatonically decreasing function of input level \cite{Braje_2003aa}, thus the output level is sensitive to variations in the input level. The signal level will not undergo any correction or restoration in passing from one device to another.

Other low-light-level all-optical switching experiments have also been demonstrated recently in traveling-wave systems. By modifying the correlation between down-converted photons, Resch \etal \cite{Resch_2002aa} created a conditional-phase switch that operates at the single photon level. Using six-wave mixing in cold atoms, Kang \etal \cite{Kang_2004aa} demonstrated optical control of a field with 0.2 photons/$\sigma$ with a 2~photon/$\sigma$ input switching field ($\sim$$10^8$ input photons, over $\sim$0.54~$\mu$s in a $\sim$0.5~mm diameter beam). Both of these results exhibit high sensitivity but are limited to control fields that are stronger than the output field.

Another approach combines the field enhancement offered by optical cavities with the strong coupling of coherently prepared atoms. Bistability in the output of a cavity filled with a large-Kerr, EIT medium \cite{Wang_2002aa} exhibits switching that is sensitive to $~0.4$~mW of input power over a few microseconds ($\sim$$10^9$ total photons or $\sim$$10^8$~photons/$\sigma$, 80~$\mu$m radius, $\sim$2~$\mu$s response time). Photonic crystal nanocavities have also shown bistability switching. Tanabe \etal \cite{Tanabe_2005aa} demonstrated switching with 74~fJ pulses and a switching speed of $<100$~ps ($\sim$500,000 photons) in a Silicon photonic crystal nanocavity system. There has also been a recent proposal to use photonic crystal microcavities filled with an ultra-slow-light EIT medium \cite{Soljacic_2005aa} as a switching platform. Simulations for such a system suggest switching could be achieved with less than 3000 photons\footnote{Because the system has already been reduced to the minimum possible transverse dimension, the mode volume of the proposed photonic crystal microcavity is less than $\lambda^3$ and the photons/$\sigma$ metric is no longer applicable to photonic crystal systems.}. Taking a different approach, Islam \etal \cite{Islam_1988aa} exploit a modulational instability in an optical fiber interferometer to gate the transmission of 184~mW by injecting only 4.4~$\mu$W, which corresponds to injecting less than 2,000 photons during the 50~psec switching time. Furthermore, with an effective area of 2.6$\times10^{-7}$~cm$^2$, this sensitivity corresponds to 24~photons/$\sigma$.

Many of these other systems satisfy some, but not all, of the criteria for scalability. Of the two most sensitive systems just discussed, EIT-filled photonic crystal microcavities, suffers from the same drawbacks as cold-atom EIT systems: the input and output fields are required to have the same power, making them not cascadable. The other highly sensitive system, a modulational-instability fiber interferometer, is both cascadable and exhibits signal level restoration. In several ways the latter system is similar to ours: it exploits the sensitivity of instabilities and uses a sensitive detector (in their case the interferometer, in my case pattern orientation) to distinguish states of the switch.

Finally, there has been a very recent proposal that does not use cavities or traveling optical fields, but instead takes advantage of photon-induced surface plasmons excited in a conducting nano-wire that couple strongly to a two-level emitter placed nearby. This strong coupling enables effects that are similar to those observed in cavity QED. Specifically, Chang \etal \cite{Chang_2007aa} suggest that a system consisting of a nano-wire coupled to a dielectric waveguide could be used to create an optical transistor that is sensitive to a single photon \cite{Chang_2007aa}. Photons in the dielectric waveguide are efficiently coupled to plasmons that propagate along the nanowire. A two-level emitter placed close to the nanowire has a strong effect on the plasmon transmission. The absorption of a single photon by the emitter is sufficient to change the nanowire from complete plasmon reflection to complete plasmon transmission.

This system is very similar in effect to cavity QED systems, with the added advantage that the input and output are separate modes and thus separate channels. There are certainly experimental challenges to implementing this proposal but there do not appear to be any fundamental limitations to its application in an optical network. If implemented as proposed, a surface-plasmon transistor could operate with single-photon input levels, and gate signals containing many photons. The strength of the gated signal is limited by the effective Purcell factor, which depends on the nano-wire diameter, and ranges from 1 to 1000 for nanowire diameters in the range of 100 to 3 nm \cite{Chang_2007aa}. Hence the one-photon transistor could control a signal containing up to 1000 photons, indicating such a device would be cascadable.

Many all-optical switches have been successfully demonstrated over a period spanning several decades. However, in almost every case, one or more important features is missing from the switching device. With the requirements of scalability and sensitivity in mind, this thesis reports on a new approach to all-optical switching.

\subsection{Switching with Transverse Optical Patterns}

My approach to all-optical switching is to exploit collective instabilities that occur when laser beams interact with a nonlinear medium \cite{Lugiato_1994aa}. One such collective instability occurs when laser beams counterpropagate through an atomic vapor. In this configuration, given sufficiently strong nonlinear interaction strength, it is known that mirror-less parametric self-oscillation gives rise to stationary, periodic, or chaotic behavior of the intensity \cite{Silberberg_1984aa,Khitrova_1988aa} and/or polarization \cite{Gaeta_1987aa,Gauthier_1988aa,Gauthier_1990aa}.

Another well-known feature of counterpropagating beam instabilities is the formation of transverse optical patterns, \emph{i.e.}, the formation of spatial structure of the electromagnetic field in the plane perpendicular to the propagation direction \cite{Petrossian_1992aa,Lugiato_1994aa}. This is also true for my experiment where a wide variety of patterns can be generated, including rings and multi-spot off-axis patterns in agreement with previous experiments \cite{Petrossian_1992aa,Grynberg_1988aa,Gauthier_1990aa}.

Building an all-optical switch from transverse optical patterns combines several well-known features of nonlinear optics in a novel way. Near-resonance enhancement of the atom-photon coupling makes my system sensitive to weak optical fields. Using optical fields with a counterpropagating beam geometry allows for interactions with atoms in specific velocity groups leading to  sub-Doppler nonlinear optics without requiring cold atoms. Finally, using the different orientations of a transverse pattern as distinct states of a switch  allows me to maximize the sensitivity of the pattern forming instability. Instabilities, by nature, are sensitive to perturbations, so by combining instabilities with resonantly-enhanced, sub-Doppler nonlinearities I created a switch with very high sensitivity.

Before describing my experimental setup and results, it is important to include a more complete background for the treatment of optical interactions in nonlinear media. Specifically, there are several fundamental concepts that are important to understanding the origin of transverse pattern formation in nonlinear optics. The following chapter presents a basic model for the interaction between light waves in nonlinear media.


